Multi-resonant coupling architectures for ZZ interaction reduction

ABSTRACT

Systems and techniques that facilitate multi-resonant couplers for preserving ZX interaction while reducing ZZ interaction are provided. In various embodiments, a first qubit can have a first operational frequency and a second qubit can have a second operational frequency, and a multi-resonant architecture can couple the first qubit to the second qubit. In various embodiments, the multi-resonant architecture can comprise a first resonator and a second resonator. In various cases, the first resonator can capacitively couple the first qubit to the second qubit, and a second resonator can capacitively couple the first qubit to the second qubit. In various aspects, the first resonator and the second resonator can be in parallel. In various instances, the first resonator can have a first resonant frequency less than the first operational frequency and the second operational frequency, and the second resonator can have a second resonant frequency greater than the first operational frequency and the second operational frequency. In various other embodiments, the multi-resonant architecture can comprise a resonator, a first end of which can be capacitively coupled to the first qubit and to the second qubit, and a second end of which can be coupled to ground. In various instances, the resonator can have a first harmonic less than the first operational frequency and the second operational frequency, and can have a second harmonic greater than the first operational frequency and the second operational frequency. In various other embodiments, the multi-resonant architecture can comprise a resonator and a direct coupler. In various embodiments, the resonator and the direct coupler can both capacitively couple the first qubit to the second qubit, and the resonator and the direct coupler can be in parallel. In various cases, the direct coupler can couple opposite pads of the first qubit and the second qubit. In various embodiments, a first end of the resonator can be capacitively coupled to the first qubit and the second qubit, a second end of the resonator can be coupled to ground, and the direct coupler can capacitively couple common pads of the first qubit and the second qubit.

BACKGROUND

The subject disclosure relates generally to superconducting qubits, andmore specifically to multi-resonant coupling architectures that reduceZZ interaction between superconducting qubits while enabling crossresonance gates via the ZX interaction.

Quantum computing systems can be composed of various arrangements ofsuperconducting qubits. In various instances, the qubits can have fixedoperational frequencies (e.g., a transmon qubit with a single Josephsonjunction can have a fixed operational frequency) and can be arranged intwo-dimensional arrays on any suitable quantum computing substrate. Invarious aspects, any qubit in such a two-dimensional array can becoupled to some and/or all of its nearest-neighbor qubits and/or to someand/or all of its next-nearest neighbor qubits.

Conventionally, qubits are coupled together via fixed-frequencymicrowave resonators (e.g., bus resonators). That is, a first qubit anda second qubit are conventionally coupled by a single fixed-frequencyresonator, where a first end of the single fixed-frequency resonator iscapacitively coupled to the first qubit, and where a second end of thesingle fixed-frequency resonator is capacitively coupled to the secondqubit. Such a coupling allows the first qubit and the second qubit toexhibit high coherence and/or a strong ZX interaction from crossresonance, which can improve the functioning of the overall quantumcomputing system. In various instances, qubit devices comprising morethan 50 qubits have been successfully implemented based on suchcross-resonance interactions, where the qubits are driven with microwavetones at the frequency of neighboring qubits.

A significant disadvantage of conventional couplers, however, is thatthey result in an always-on ZZ interaction between the coupled qubits.This weak ZZ error accumulates between any conventionally coupled pairof qubits and corrodes the desired cross resonance mechanism used fortwo-qubit gates. In other words, this ZZ error inhibits theeffectiveness and/or efficacy of quantum computing systems. Conventionalsystems and/or techniques for dealing with the always-on ZZ errorinclude echoing and tunable-frequency coupling elements. Echoinginvolves using additional pulses to cancel the ZZ interaction. Thesepulses, however, require time to implement, which can significantly eatinto the coherence budget due to finite coherence times.Tunable-frequency couplers can be used to reduce and/or eliminate the ZZinteraction. However, adding tunable-frequency elements to quantumcomputing systems often results in coherence degradation. In otherwords, conventional systems and/or techniques for reducing the always-onZZ interaction have corresponding negative impacts on coherence times.

In various instances, embodiments of the invention can solve one or moreof these problems in the prior art.

SUMMARY

The following presents a summary to provide a basic understanding of oneor more embodiments of the invention. This summary is not intended toidentify key or critical elements, or delineate any scope of theparticular embodiments or any scope of the claims. Its sole purpose isto present concepts in a simplified form as a prelude to the moredetailed description that is presented later. In one or more embodimentsdescribed herein, devices, systems, computer-implemented methods,apparatus and/or computer program products that facilitatemulti-resonant coupling architectures for ZZ interaction reduction aredescribed.

According to one or more embodiments, a device is provided. The devicecan comprise a first qubit, a second qubit, and a multi-resonantarchitecture. In various aspects, the multi-resonant architecture cancomprise a first resonator that capacitively couples the first qubit tothe second qubit and a second resonator that capacitively couples thefirst qubit to the second qubit. In various embodiments, the first qubitcan have a first operational frequency, and the second qubit can have asecond operational frequency. In various cases, the first resonator canhave a first resonant frequency that is less than the first operationalfrequency and that is less than the second operational frequency. Invarious aspects, the second resonator can have a second resonantfrequency that is greater than the first operational frequency and thatis greater than the second operational frequency. In variousembodiments, the first resonator and the second resonator can be λ/2resonators, and the first resonator and the second resonator can be inparallel. In various embodiments, the first resonant frequency can beabout 3 gigahertz (GHz), the second resonant frequency can be about 6GHz, and the first operational frequency and the second operationalfrequency can be between 4.5 GHz and 5.5 GHz. In various instances, thefirst resonant frequency, the second resonant frequency, the firstoperational frequency, and the second operational frequency can befixed.

According to one or more embodiments, a device is provided. The devicecan comprise a first qubit, a second qubit, and a multi-resonantarchitecture. In various aspects, the multi-resonant architecture cancomprise a resonator. In various instances, a first end of the resonatorcan be capacitively coupled to the first qubit and to the second qubit.In various aspects, a second end of the resonator can be coupled toground. In various embodiments, the first qubit can have a firstoperational frequency, and the second qubit can have a secondoperational frequency. In various cases, the resonator can have a firstharmonic frequency that is less than the first operational frequency andthat is less than the second operational frequency. In various aspects,the resonator can have a second harmonic frequency that is greater thanthe first operational frequency and that is greater than the secondoperational frequency. In various embodiments, the resonator can be aλ/4 resonator. In various embodiments, the first harmonic frequency canbe about 2 gigahertz (GHz), the second harmonic frequency can be about 6GHz, and the first operational frequency and the second operationalfrequency can be between 4.5 GHz and 5.5 GHz. In various instances, thefirst harmonic frequency, the second harmonic frequency, the firstoperational frequency, and the second operational frequency can befixed.

According to one or more embodiments, a device is provided. The devicecan comprise a first qubit, a second qubit, and a multi-resonantarchitecture. In various aspects, the multi-resonant architecture cancomprise a resonator and a differential direct coupler. In variousinstances, the resonator can capacitively couple the first qubit to thesecond qubit, and the differential direct coupler can capacitivelycouple the first qubit to the second qubit. In various cases, thedifferential direct coupler can capacitively couple opposite pads of thefirst qubit and the second qubit. In various embodiments, the firstqubit can have a first operational frequency, and the second qubit canhave a second operational frequency. In various cases, the resonator canhave a resonant frequency that is greater than the first operationalfrequency and that is greater than the second operational frequency. Invarious embodiments, the resonator can be a λ/2 resonator, and theresonator and the differential direct coupler can be in parallel. Invarious embodiments, the resonant frequency can be about 6 gigahertz(GHz), and the first operational frequency and the second operationalfrequency can be between 4.5 GHz and 5.5 GHz. In various instances, theresonant frequency, the first operational frequency, and the secondoperational frequency can be fixed.

According to one or more embodiments, a device is provided. The devicecan comprise a first qubit, a second qubit, and a multi-resonantarchitecture. In various aspects, the multi-resonant architecture cancomprise a resonator and a direct coupler. In various instances, a firstend of the resonator can be capacitively coupled to the first qubit andto the second qubit, and a second end of the resonator can be coupled toground. In various aspects, the direct coupler can capacitively couplethe first qubit to the second qubit. In various cases, the directcoupler can capacitively couple common pads of the first qubit and thesecond qubit. In various embodiments, the first qubit can have a firstoperational frequency, and the second qubit can have a secondoperational frequency. In various cases, the resonator can have aresonant frequency that is greater than the first operational frequencyand that is greater than the second operational frequency. In variousembodiments, the resonator can be a λ/4 resonator. In variousembodiments, the resonant frequency can be about 6 gigahertz (GHz), andthe first operational frequency and the second operational frequency canbe between 4.5 GHz and 5.5 GHz. In various instances, the resonantfrequency, the first operational frequency, and the second operationalfrequency can be fixed.

According to one or more embodiments, an apparatus is provided. Theapparatus can comprise a first transmon qubit having a first operationalfrequency, a second transmon qubit having a second operationalfrequency, and a multi-resonant architecture. In various aspects, themulti-resonant architecture can capacitively couple the first transmonqubit to the second transmon qubit. In various instances, themulti-resonant architecture can have a first resonant frequency that isless than the first operational frequency and that is less than thesecond operational frequency, and can have a second resonant frequencythat is greater than the first operational frequency and that is greaterthan the second operational frequency. In various embodiments, themulti-resonant architecture can comprise a first λ/2 resonatorcapacitively coupled to the first transmon qubit and to the secondtransmon qubit, wherein the first λ/2 resonator exhibits the firstresonant frequency. In various instances, the multi-resonantarchitecture can comprise a second λ/2 resonator capacitively coupled tothe first transmon qubit and to the second transmon qubit, wherein thesecond λ/2 resonator exhibits the second resonant frequency. In variouscases, the first λ/2 resonator and the second λ/2 resonator can be inparallel. In various other embodiments, the multi-resonant architecturecan comprise a λ/4 resonator. In various instances, a first end of theλ/4 resonator can be coupled between coupling capacitors of the firsttransmon qubit and the second transmon qubit, and a second end of theλ/4 resonator can be shorted to ground. In various cases, a firstharmonic of the λ/4 resonator can be the first resonant frequency, and asecond harmonic of the λ/4 resonator can be the second resonantfrequency.

As mentioned above, a pair of fixed-frequency qubits are conventionallycoupled together via a fixed-frequency microwave resonator.Specifically, for a first qubit and a second qubit, a first end of thefixed-frequency microwave resonator is capacitively coupled to the firstqubit and a second end of the fixed-frequency microwave resonator iscapacitively coupled to the second qubit. Such a coupling structure canresult in high coherence and/or strong ZX interaction between the firstqubit and the second qubit. However, such a coupling structure alsogenerates an always-on ZZ interaction between the first qubit and thesecond qubit. This ZZ interaction can negatively affect the performanceof a quantum computing system that includes the first qubit and thesecond qubit. Thus, elimination, minimization, suppression, and/orreduction of this ZZ interaction can improve the functioning of thequantum computing system.

As explained above, there are two main conventional systems and/ortechniques for suppressing and/or reducing ZZ interactions. The firstconventional system and/or technique is echoing. Echoing involvesinjecting additional pulses into the quantum computing system tocounteract, cancel, and/or destructively interfere with the ZZinteractions. However, injecting these pulses requires time, and thetime spent injecting these pulses can eat into the coherence budget ofthe quantum computing system. The second conventional system and/ortechnique for dealing with ZZ interactions is to use tunable-frequencyelements. Introducing tunable-frequency elements into the quantumcomputing system can eliminate and/or reduce the ZZ interaction.However, the use and/or complexity of tunable-frequency elementsintroduces a corresponding coherence degradation. In other words,conventional systems and/or techniques reduce ZZ interactions between acoupled pair of qubits at the cost of decreased coherence times.

Various embodiments of the invention can solve one or more of theseproblems in the prior art. In various aspects, embodiments of theinvention can provide a multi-resonant coupling architecture that cancouple a first qubit to a second qubit. In various instances, such amulti-resonant coupling architecture can reduce the ZZ interactionbetween the first qubit and the second qubit without reducing thecoupling strength and/or the ZX interaction between the first qubit andthe second qubit. In various instances, such a multi-resonant couplingarchitecture can comprise fixed-frequency and/or non-tunable elements,and so such a multi-resonant coupling architecture can avoid introducinginto a quantum computing system the coherence degradation that normallyaccompanies tunable-frequency elements. Moreover, such a multi-resonantcoupling architecture can, in various aspects, dispense with the need toinject echoes into the quantum computing system. In other words, variousembodiments of the invention can provide multi-resonant couplingarchitectures that can reduce ZZ interactions between coupled qubitswithout introducing a corresponding decrease in coherence times, unlikeconventional systems and/or techniques.

Various multi-resonant coupling architectures can be implemented toachieve these improved results. Consider a first qubit having a firstoperational frequency and a second qubit having a second operationalfrequency. In some embodiments, the multi-resonant coupling architecturecan include a first λ/2 resonator and a second λ/2 resonator. In variousinstances, a first end of the first λ/2 resonator can couple to a firstcoupling capacitor of the first qubit, and a second end of the first λ/2resonator can couple to a first coupling capacitor of the second qubit.Similarly, a first end of the second λ/2 resonator can couple to asecond coupling capacitor of the first qubit, and a second end of thesecond λ/2 resonator can couple to a second coupling capacitor of thesecond qubit. In other words, the first λ/2 resonator can capacitivelycouple the first qubit to the second qubit, and the second λ/2 resonatorcan capacitively couple the first qubit to the second qubit, such thatthe first λ/2 resonator and the second λ/2 resonator are in parallel. Invarious instances, the first λ/2 resonator can exhibit a first resonantfrequency that is less than the first operational frequency and that isless than the second operational frequency. In various aspects, thesecond λ/2 resonator can exhibit a second resonant frequency that isgreater than the first operational frequency and that is greater thanthe second operational frequency. Moreover, in various aspects, thefirst resonant frequency and the second resonant frequency can be fixed.In various embodiments, such a multi-resonant coupling architecture canreduce (e.g., in some cases, by an order of magnitude and/or more) ZZinteraction between the first qubit and the second qubit withoutcorrespondingly reducing the coupling strength and/or thecross-resonance gate speed between the first qubit and the second qubit.Moreover, such a multi-resonant coupling architecture can avoidintroducing coherence degradation (e.g., echoes and/or tunable-frequencyelements can be not required).

In other embodiments, the multi-resonant coupling architecture caninclude a λ/4 resonator. In various aspects, a first end of the λ/4resonator can couple to a coupling capacitor of the first qubit, and thefirst end of the λ/4 resonator can also couple to a coupling capacitorof the second qubit. That is, in various aspects, the first end of theλ/4 resonator can be capacitively coupled to both the first qubit andthe second qubit. In various instances, a second end of the λ/4resonator can be coupled to ground. In various aspects, the λ/4resonator can exhibit a first harmonic frequency that is less than thefirst operational frequency and that is less than the second operationalfrequency. In various instances, the λ/4 resonator can exhibit a secondharmonic frequency that is greater than the first operational frequencyand that is greater than the second operational frequency. Moreover, invarious aspects, the first harmonic frequency and the second harmonicfrequency can be fixed. In various embodiments, such a multi-resonantcoupling architecture can reduce (e.g., in some cases, by an order ofmagnitude and/or more) ZZ interaction between the first qubit and thesecond qubit without correspondingly reducing the coupling strengthand/or cross-resonance gate speed between the first qubit and the secondqubit. Moreover, such a multi-resonant coupling architecture can avoidintroducing coherence degradation (e.g., echoes and/or tunable-frequencyelements can be not required).

In still other embodiments, the multi-resonant coupling architecture caninclude a λ/2 resonator and a differential direct coupler. In variousinstances, a first end of the λ/2 resonator can couple to a firstcoupling capacitor of the first qubit, and a second end of the λ/2resonator can couple to a first coupling capacitor of the second qubit.Similarly, a first end of the differential direct coupler can couple toa second coupling capacitor of the first qubit, and a second end of thedifferential direct coupler can couple to a second coupling capacitor ofthe second qubit. In other words, the λ/2 resonator can capacitivelycouple the first qubit to the second qubit, and the differential directcoupler can capacitively couple the first qubit to the second qubit,such that the λ/2 resonator and the differential direct coupler are inparallel. In various aspects, the differential direct coupler can coupleopposite pads of the first qubit and the second qubit. In variousinstances, the λ/2 resonator can exhibit a resonant frequency that isgreater than the first operational frequency and that is greater thanthe second operational frequency. Moreover, in various aspects, theresonant frequency can be fixed. In various embodiments, such amulti-resonant coupling architecture can reduce (e.g., in some cases, byan order of magnitude and/or more) ZZ interaction between the firstqubit and the second qubit without correspondingly reducing the couplingstrength and/or cross-resonance gate speed between the first qubit andthe second qubit. Moreover, such a multi-resonant coupling architecturecan avoid introducing coherence degradation (e.g., echoes and/ortunable-frequency elements can be not required).

In yet other embodiments, the multi-resonant coupling architecture caninclude a λ/4 resonator and a direct coupler. In various aspects, afirst end of the λ/4 resonator can couple to a first coupling capacitorof the first qubit, and the first end of the λ/4 resonator can alsocouple to a first coupling capacitor of the second qubit. That is, invarious aspects, the first end of the λ/4 resonator can be capacitivelycoupled to both the first qubit and the second qubit. In variousinstances, a second end of the λ/4 resonator can be coupled to ground.In various cases, a first end of the direct coupler can couple to asecond coupling capacitor of the first qubit, and a second end of thedirect coupler can couple to a second coupling capacitor of the secondqubit. In various cases, the direct coupler can couple common pads ofthe first qubit and the second qubit. In various aspects, the λ/4resonator can exhibit a resonant frequency that is greater than thefirst operational frequency and that is greater than the secondoperational frequency. Moreover, in various aspects, the resonantfrequency can be fixed. In various embodiments, such a multi-resonantcoupling architecture can reduce (e.g., in some cases, by an order ofmagnitude and/or more) ZZ interaction between the first qubit and thesecond qubit without correspondingly reducing the coupling strengthand/or cross-resonance gate speed between the first qubit and the secondqubit. Moreover, such a multi-resonant coupling architecture can avoidintroducing coherence degradation (e.g., echoes and/or tunable-frequencyelements can be not required).

Therefore, various embodiments of the invention can providemulti-resonant coupling architectures that can reduce ZZ interactionsbetween coupled qubits without correspondingly reducing coherence times,unlike conventional systems and/or techniques. Thus, various embodimentsof the invention constitute a concrete technical improvement over theprior art.

DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a block diagram of an example, non-limiting systemincluding two resonators that facilitates ZZ interaction reduction inaccordance with one or more embodiments described herein.

FIG. 2 illustrates a block diagram of an example, non-limiting systemincluding one resonator that facilitates ZZ interaction reduction inaccordance with one or more embodiments described herein.

FIG. 3 illustrates a block diagram of an example, non-limiting systemincluding a resonator and a differential direct coupler that facilitatesZZ interaction reduction in accordance with one or more embodimentsdescribed herein.

FIG. 4 illustrates a block diagram of an example, non-limiting systemincluding a resonator and a direct coupler that facilitates ZZinteraction reduction in accordance with one or more embodimentsdescribed herein.

FIGS. 5-6 illustrate example, non-limiting graphs that depict the ZZinteraction reduction facilitated by one or more embodiments describedherein.

FIG. 7 illustrates a block diagram of an example, non-limiting qubitarray that facilitates ZZ interaction reduction in accordance with oneor more embodiments described herein.

FIG. 8 illustrates a flow diagram of an example, non-limiting methodincluding two resonators that facilitates ZZ interaction reduction inaccordance with one or more embodiments described herein.

FIG. 9 illustrates a flow diagram of an example, non-limiting methodincluding one resonator that facilitates ZZ interaction reduction inaccordance with one or more embodiments described herein.

FIG. 10 illustrates a flow diagram of an example, non-limiting methodincluding a resonator and a differential direct coupler that facilitatesZZ interaction reduction in accordance with one or more embodimentsdescribed herein.

FIG. 11 illustrates a flow diagram of an example, non-limiting methodincluding a resonator and a direct coupler that facilitates ZZinteraction reduction in accordance with one or more embodimentsdescribed herein.

FIG. 12 illustrates a flow diagram of an example, non-limiting methodthat facilitates ZZ interaction reduction in accordance with one or moreembodiments described herein.

FIG. 13 illustrates a block diagram of an example, non-limitingoperating environment in which one or more embodiments described hereincan be facilitated.

DETAILED DESCRIPTION

The following detailed description is merely illustrative and is notintended to limit embodiments and/or application or uses of embodiments.Furthermore, there is no intention to be bound by any expressed orimplied information presented in the preceding Background or Summarysections, or in the Detailed Description section.

One or more embodiments are now described with reference to thedrawings, wherein like referenced numerals are used to refer to likeelements throughout. In the following description, for purposes ofexplanation, numerous specific details are set forth in order to providea more thorough understanding of the one or more embodiments. It isevident, however, in various cases, that the one or more embodiments canbe practiced without these specific details.

Consider two transmon qubits coupled by a fixed-frequency bus resonator,as in conventional systems and/or techniques. The coupling strength canbe quantified by the exchange coupling J, which can be given by:

$J = \frac{g_{1}{g_{2}\left( {\Delta_{1} + \Delta_{2}} \right)}}{2\Delta_{1}\Delta_{2}}$where Δ₁ is the difference between the operational frequency of thefirst transmon qubit and the resonant frequency of the fixed-frequencybus resonator, where Δ₂ is the difference between the operationalfrequency of the second transmon qubit and the resonant frequency of thefixed-frequency bus resonator, where g₁ is the coupling strength of thefirst transmon qubit to the fixed-frequency bus resonator, and where g₂is the coupling strength of the second transmon qubit to thefixed-frequency bus resonator. Also consider two transmon qubits coupledby a direct coupler; that is, any suitable, short section oftransmission line (e.g., short in the sense that its resonance frequencyis greater than about 30 GHz), which can give rise to a qubitfrequency-independent coupling at typical transmon qubit frequencies ofaround 5 GHz. In this case the qubit coupling is also given by a certainvalue of J set by the direct coupler geometry. In both these instances,the Hamiltonian H of these conventionally coupled qubits (e.g., twoqubits coupled by a single fixed-frequency microwave resonator orcoupled by a direct coupler) can be described by the following equation:

$H = {{\sum\limits_{i}\left\lbrack {{\omega_{i}{\overset{\hat{}}{n}}_{i}} + {\frac{\alpha_{i}}{2}{{\overset{\hat{}}{n}}_{i}\left( {{\overset{\hat{}}{n}}_{i} - 1} \right)}}} \right\rbrack} + {J\left( {{{\overset{\hat{}}{a}}_{1}{\overset{\hat{}}{a}}_{2}^{\dagger}} + {{\overset{\hat{}}{a}}_{2}{\overset{\hat{}}{a}}_{1}^{\dagger}}} \right)}}$where i can take values 1 or 2 to represent the first or second transmonqubit, wherein ω_(i) is the resonance frequency of the i-th transmonqubit, where {circumflex over (n)}_(i) denotes the number of excitationsin the i-th transmon qubit, where α_(i) is the anharmonicity of the i-thtransmon qubit, and where â₁â₂ ^(†)+â₂â₁ ^(†) represents the exchangecoupling between qubits 1 and 2 (â_(i) is the i-th qubit annihilationoperator and â_(i) ^(†) is the i-th qubit creation operator). In thisHamiltonian, the always on ZZ interaction can be given as:

${ZZ} = \frac{2{J^{2}\left( {\alpha_{1} + \alpha_{2}} \right)}}{\left( {\Delta + \alpha_{1}} \right)\left( {\Delta - \alpha_{2}} \right)}$where Δ is the detuning between the two qubits and where all othersymbols are defined in the two above equations. This arises from thefact that the sum of the |00

and |11

state energies of the two transmon qubits is different from the sum ofthe |01

and |10

state energies. Although a low ZZ interaction between the two transmonqubits is desirable, a high coupling strength between the two transmonqubits is also desirable, since conversely the ZX interaction from qubit1 to 2 (e.g., the strength of cross-resonance) is given by:

${ZX} = {\frac{J\Omega_{1}}{\Delta}\left( \frac{\alpha_{1}}{\Delta + \alpha_{1}} \right)}$where J, α₁, and Δ are as defined above, and where Ω₁ is thecross-resonance drive strength applied to qubit 1. As shown by the aboveequations, one way to increase the ratio of ZX to ZZ (that is, toincrease the ratio of wanted to unwanted interaction) would be todecrease J. However, this increases the overall time of the gateoperation, thus leading to degradation of the fidelity due to loss ofcoherence.

As mentioned above, conventional systems and/or techniques for reducing,suppressing, eliminating, and/or minimizing the ZZ interaction withoutalso reducing, suppressing, eliminating, and/or minimizing the ZXinteraction implement echoing and/or tunable-frequency elements. Echoinginvolves injecting multiple pulse signals into a quantum computingsystem to counteract, cancel, negate, correct, and/or destructivelyinterfere with the ZZ interaction. However, time is required to injectsuch pulse signals into the quantum computing system, and that time caneat into the already-limited coherence budget of the quantum computingsystem. Tunable-frequency elements can be used to ameliorate the ZZinteraction. However, tunable-frequency elements are also associatedwith coherence degradation. Thus, conventional systems and/or techniquesreduce the ZZ interaction at the expense of decreased coherence times.Various embodiments of the invention, however, can reduce the ZZinteraction without this corresponding decrease in coherence times.

The inventors of various embodiments of the invention recognized thatthe ZZ interaction can, in various instances, be reduced and/orcancelled while maintaining a finite J by incorporating a second couplermode. In various instances, the Hamiltonian H when a second coupler modeis incorporated can be described by the following equation:

$H = {{\sum\limits_{i}\left\lbrack {{\omega_{i}{\overset{\hat{}}{n}}_{i}} + {\frac{\alpha_{i}}{2}{{\overset{\hat{}}{n}}_{i}\left( {{\overset{\hat{}}{n}}_{i} - 1} \right)}} + {\sum\limits_{j}{g_{ij}\left( {{{\overset{\hat{}}{a}}_{i}{\overset{\hat{}}{b}}_{j}^{\dagger}} + {{\overset{\hat{}}{b}}_{j}{\overset{\hat{}}{a}}_{i}^{\dagger}}} \right)}}} \right\rbrack} + {\sum\limits_{j}{\gamma_{j}{\overset{\hat{}}{b}}_{j}^{\dagger}{\overset{\hat{}}{b}}_{j}}} + {J_{0}\left( {{{\overset{\hat{}}{a}}_{1}{\overset{\hat{}}{a}}_{2}^{\dagger}} + {{\overset{\hat{}}{a}}_{2}{\overset{\hat{}}{a}}_{1}^{\dagger}}} \right)}}$where i can take values 1 or 2 to represent the first or second transmonqubit, wherein ω_(i), α_(i), {circumflex over (n)}_(i), â_(i) aredefined above, where j is summed over the number of resonator modescoupling the first transmon qubit to the second transmon qubit, whereγ_(j) is the frequency of the resonator mode, where {circumflex over(b)}_(j) ^(†){circumflex over (b)}_(j) is the number of excitations inthe resonator mode, where denotes the coupling between the i-th transmonqubit and the coupler mode j, and where (â_(i){circumflex over (b)}_(j)^(†)+{circumflex over (b)}_(j)â_(i) ^(†)) represents the exchangebetween the i-th transmon qubit and the coupler mode j. In this form,the remaining J₀ coupled term is due to a direct coupler if one exists.

Various embodiments of the invention can provide multi-resonant couplingarchitectures that can reduce the ZZ interaction between two qubitswhile maintaining a finite exchange coupling J and without acorresponding decrease in coherence times. Again, consider a first qubithaving a first operational frequency and a second qubit having a secondoperational frequency. In various aspects, a multi-resonant architecturecan capacitively couple the first qubit to the second qubit. In variousinstances, the multi-resonant architecture can have a first pole that isgreater than the first operational frequency and that is greater thanthe second operational frequency. In various instances, themulti-resonant architecture can have a second pole that is less than thefirst operational frequency and that is less than the second operationalfrequency. In various cases, rather than the second pole, themulti-resonant architecture can have a direct coupling term (e.g., adirect coupler that capacitively couples the first qubit to the secondqubit). In various instances, the multi-resonant architecture canexhibit a zero ZZ interaction and a zero exchange coupling J in a firstset of qubit frequencies. In various aspects, the multi-resonantarchitecture can exhibit a zero ZZ interaction and a non-zero exchangecoupling J in a second set of qubit frequencies. In various cases, themulti-resonant architecture can be non-tunable.

In various embodiments, the multi-resonant architecture can comprise afirst resonator and a second resonator. In various instances, the firstresonator can capacitively couple the first qubit to the second qubit.That is, a first end of the first resonator can couple to a firstcoupling capacitor of the first qubit, and a second end of the firstresonator can couple to a first coupling capacitor of the second qubit.Similarly, the second resonator can capacitively couple the first qubitto the second qubit. That is, a first end of the second resonator cancouple to a second coupling capacitor of the first qubit, and a secondend of the second resonator can couple to a second coupling capacitor ofthe second qubit. In various instances, the first resonator can be inparallel with the second resonator. In various aspects, the firstresonator and the second resonator can both be λ/2 resonators. Invarious instances, the first resonator can have a first resonantfrequency that is less than the first operational frequency of the firstqubit and that is less than the second operational frequency of thesecond qubit. In various cases, the second resonator can have a secondresonant frequency that is greater than the first operational frequencyof the first qubit and that is greater than the second operationalfrequency of the second qubit. In various instances, the first resonantfrequency can be about 3 GHz, the second resonant frequency can be about6 GHz, and the first operational frequency and the second operationalfrequency can be in the range of 4.5 GHz to 5.5 GHz. In variousinstances, the first resonator and/or the second resonator can benon-tunable. In various aspects, such a multi-resonant architecture canreduce (e.g., in some cases, by an order of magnitude and/or more) theZZ interaction between the first qubit and the second qubit withoutcorrespondingly decreasing the ZX interaction and/or the exchangecoupling J between the first qubit and the second qubit. Moreover, sucha multi-resonant architecture can accomplish this result withoutimplementing many-pulse echoes and/or without tunable-frequencyelements. Thus, such a multi-resonant architecture can, in variousinstances, reduce the ZZ interaction without the coherence degradationthat accompanies conventional systems and/or techniques.

In various other embodiments, the multi-resonant architecture cancomprise a resonator. In various instances, a first end of the resonatorcan be capacitively coupled to a coupling capacitor of the first qubitand can be capacitively coupled to a coupling capacitor of the secondqubit. That is, the first end of the resonator can be capacitivelycoupled to both the first qubit and the second qubit. In variousinstances, a second end of the resonator can be coupled to ground. Invarious aspects, the resonator can be a λ/4 resonator. In variousinstances, the resonator can have a first harmonic frequency that isless than the first operational frequency of the first qubit and that isless than the second operational frequency of the second qubit. Invarious cases, the resonator can have a second harmonic frequency thatis greater than the first operational frequency of the first qubit andthat is greater than the second operational frequency of the secondqubit. In various instances, the first harmonic frequency can be about 2GHz, the second harmonic frequency can be about 6 GHz, and the firstoperational frequency and the second operational frequency can be in therange of 4.5 GHz to 5.5 GHz. In various instances, the resonator can benon-tunable. In various aspects, such a multi-resonant architecture canreduce (e.g., in some cases, by an order of magnitude and/or more) theZZ interaction between the first qubit and the second qubit withoutcorrespondingly decreasing the ZX interaction and/or the exchangecoupling J between the first qubit and the second qubit. Moreover, sucha multi-resonant architecture can accomplish this result withoutimplementing many-pulse echoes and/or without tunable-frequencyelements. Thus, such a multi-resonant architecture can, in variousinstances, reduce the ZZ interaction without the coherence degradationthat accompanies conventional systems and/or techniques.

In various other embodiments, the multi-resonant architecture cancomprise a resonator and a differential direct coupler. In variousinstances, the resonator can capacitively couple the first qubit to thesecond qubit. That is, a first end of the resonator can couple to afirst coupling capacitor of the first qubit, and a second end of theresonator can couple to a first coupling capacitor of the second qubit.Similarly, the differential direct coupler can, in various instances,capacitively couple the first qubit to the second qubit. That is, afirst end of the differential direct coupler can couple to a secondcoupling capacitor of the first qubit, and a second end of thedifferential direct coupler can couple to a second coupling capacitor ofthe second qubit. In various instances, the differential direct couplercan couple together opposite pads of the first qubit and the secondqubit. In various cases, the resonator can be in parallel with thedifferential direct coupler. In various aspects, the resonator can be aλ/2 resonator. In various instances, the resonator can have a resonantfrequency that is greater than the first operational frequency of thefirst qubit and that is greater than the second operational frequency ofthe second qubit. In various instances, the resonant frequency can beabout 6 GHz, and the first operational frequency and the secondoperational frequency can be in the range of 4.5 GHz to 5.5 GHz. Invarious instances, the resonator can be non-tunable. In various aspects,the differential direct coupler can be any suitable, short section oftransmission line (e.g., short in the sense that its resonance frequencyis greater than about 30 GHz), which can give rise to afrequency-independent coupling at typical transmon qubit frequencies ofaround 5 GHz. In various aspects, such a multi-resonant architecture canreduce (e.g., in some cases, by an order of magnitude and/or more) theZZ interaction between the first qubit and the second qubit withoutcorrespondingly decreasing the ZX interaction and/or the exchangecoupling J between the first qubit and the second qubit. Moreover, sucha multi-resonant architecture can accomplish this result withoutimplementing many-pulse echoes and/or without tunable-frequencyelements. Thus, such a multi-resonant architecture can, in variousinstances, reduce the ZZ interaction without the coherence degradationthat accompanies conventional systems and/or techniques.

In various other embodiments, the multi-resonant architecture cancomprise a resonator and a direct coupler. In various instances, a firstend of the resonator can be capacitively coupled to both the first qubitand to the second qubit. That is, a first end of the resonator cancouple to a first coupling capacitor of the first qubit, and the firstend of the resonator can also couple to a first coupling capacitor ofthe second qubit. In various aspects, the direct coupler cancapacitively couple the first qubit to the second qubit. That is, afirst end of the direct coupler can couple to a second couplingcapacitor of the first qubit, and a second end of the direct coupler cancouple to a second coupling capacitor of the second qubit. In variousinstances, the direct coupler can couple together common pads of thefirst qubit and the second qubit. In various aspects, the resonator canbe a λ/4 resonator. In various instances, the resonator can have aresonant frequency that is greater than the first operational frequencyof the first qubit and that is greater than the second operationalfrequency of the second qubit. In various instances, the resonantfrequency can be about 6 GHz, and the first operational frequency andthe second operational frequency can be in the range of 4.5 GHz to 5.5GHz. In various instances, the resonator can be non-tunable. In variousaspects, the direct coupler can be any suitable, short section oftransmission line (e.g., short in the sense that its resonance frequencyis greater than about 30 GHz), which can give rise to afrequency-independent coupling at typical transmon qubit frequencies ofaround 5 GHz. In various aspects, such a multi-resonant architecture canreduce (e.g., in some cases, by an order of magnitude and/or more) theZZ interaction between the first qubit and the second qubit withoutcorrespondingly decreasing the ZX interaction and/or the exchangecoupling J between the first qubit and the second qubit. Moreover, sucha multi-resonant architecture can accomplish this result withoutimplementing many-pulse echoes and/or without tunable-frequencyelements. Thus, such a multi-resonant architecture can, in variousinstances, reduce the ZZ interaction without the coherence degradationthat accompanies conventional systems and/or techniques.

Various embodiments of the invention include novel systems and/ortechniques for facilitating multi-resonant coupling architectures for ZZinteraction reduction that are not abstract, that are not naturalphenomena, that are not laws of nature, and that cannot be performed asa set of mental acts by a human. Instead, various embodiments of theinvention include systems and/or techniques for facilitating ZZinteraction reduction that do not correspondingly reduce ZX interactionand/or exchange coupling J, that do not require many-pulse echoes,and/or that do not require tunable-frequency elements. Reduction of ZXinteractions can negatively affect performance of a quantum computingsystem. Additionally, implementing echoes and/or tunable-frequencyelements can negatively affect the coherence budget of a quantumcomputing system. Since various embodiments of the invention can reduceZZ interactions without correspondingly reducing ZX interactions andwithout implementing echoes and/or tunable-frequency elements, variousembodiments of the invention can reduce unwanted ZZ interactions whilemaintaining a non-zero exchange coupling J without the coherencedegradation that normally accompanies conventional systems and/ortechniques. In other words, embodiments of the invention provide fornovel qubit-coupling architectures that can be implemented in quantumcomputing systems (e.g., on quantum computing chips/substrates) in orderto improve the performance and/or functioning of the quantum computingsystems. Therefore, various embodiments of the invention constituteconcrete technical improvements over the prior art.

In various aspects, it should be appreciated that the figures of thisdisclosure are exemplary and non-limiting only and are not necessarilydrawn to scale.

FIG. 1 illustrates a block diagram of an example, non-limiting system100 including two resonators that can facilitate ZZ interactionreduction in accordance with one or more embodiments described herein.As shown, in various aspects, the system 100 can comprise a first qubit102 and a second qubit 104. As shown in FIG. 1 , the first qubit 102 canbe a fixed-frequency transmon qubit. That is, the first qubit 102 cancomprise a Josephson junction 118 that is shunted by a capacitor 120. Invarious instances, however, the first qubit 102 can be any suitable typeof superconducting qubit (e.g., charge qubit, phase qubit, flux qubit).In various aspects, the first qubit 102 can be any suitablefixed-frequency superconducting qubit (e.g., a qubit whose operationalfrequency is not tunable). As also shown in FIG. 1 , the second qubit104 can be a fixed-frequency transmon qubit. That is, the second qubit104 can comprise a Josephson junction 122 that is shunted by a capacitor124. In various instances, however, the second qubit 104 can be anysuitable type of superconducting qubit (e.g., charge qubit, phase qubit,flux qubit). In various aspects, the second qubit 104 can be anysuitable fixed-frequency superconducting qubit (e.g., a qubit whoseoperational frequency is not tunable).

In various other embodiments, the first qubit 102 and/or the secondqubit 104 can be tunable and/or weakly tunable.

In various embodiments, the first qubit 102 can have a first operationalfrequency. In various instances, the second qubit 104 can have a secondoperational frequency. In various aspects, the first operationalfrequency can have any suitable value, and the second operationalfrequency can have any suitable value. In various instances, the firstoperational frequency can be within the range from 4.5 GHz to 5.5 GHz.In various aspects, the second operational frequency can be within therange of 4.5 GHz to 5.5 GHz. In various instances, the first operationalfrequency and the second operational frequency can be about 150 megaHertz (MHz) apart (e.g., a detuning and/or frequency separation of 150MHz, as measured within any suitable measurement resolution and/ormeasurement error). For example, the first operational frequency can beabout 150 MHz less than the second operational frequency. In variousembodiments, the first qubit 102 can have any suitable anharmonicity,the second qubit 104 can have any suitable anharmonicity, and the firstqubit 102 and the second qubit 104 can be in a straddling regime, wheretheir frequency separation is smaller than both qubits' anharmonicities.

In various embodiments, the first qubit 102 can have a first couplingcapacitor 108 and a second coupling capacitor 114. Similarly, the secondqubit 104 can have a first coupling capacitor 110 and a second couplingcapacitor 116. In various instances, the coupling capacitors 108, 110,114, and 116 can be any suitable coupling capacitors used in quantumcomputing systems.

In various instances, the system 100 can comprise a first resonator 106and a second resonator 112. In various aspects, the first resonator 106can be any suitable fixed-frequency microwave resonator used in quantumcomputing systems (e.g., a bus resonator). In various aspects, the firstresonator 106 can be any suitable λ/2 resonator. Similarly, in variousinstances, the second resonator 112 can be any suitable fixed-frequencymicrowave resonator used in quantum computing systems (e.g., a busresonator). In various aspects, the second resonator 112 can be anysuitable λ/2 resonator.

As shown, the first resonator 106 can, in various embodiments,capacitively couple the first qubit 102 to the second qubit 104.Specifically, in various instances, the first resonator 106 can have afirst end (e.g., the left-hand end of the first resonator 106 asdepicted in FIG. 1 ) and a second end (e.g., the right-hand end of thefirst resonator 106 as depicted in FIG. 1 ). In various cases, the firstend of the first resonator 106 can be coupled to the first couplingcapacitor 108 of the first qubit 102. In various aspects, the second endof the first resonator 106 can be coupled to the first couplingcapacitor 110 of the second qubit 104. Similarly, the second resonator112 can, in various embodiments, capacitively couple to the first qubit102 to the second qubit 104. Specifically, in various instances, thesecond resonator 112 can have a first end (e.g., the left-hand end ofthe second resonator 112 as depicted in FIG. 1 ) and a second end (e.g.,the right-hand end of the second resonator 112 as depicted in FIG. 1 ).In various cases, the first end of the second resonator 112 can becoupled to the second coupling capacitor 114 of the first qubit 102. Invarious aspects, the second end of the second resonator 112 can becoupled to the second coupling capacitor 116 of the second qubit 104.

As shown, in various instances, the first resonator 106 and the secondresonator 112 can be in parallel (e.g., as opposed to in series).

In various embodiments, the first resonator 106 can have a firstresonant frequency. In various cases, the first resonant frequency canbe less than the first operational frequency of the first qubit 102. Invarious instances, the first resonant frequency can also be less thanthe second operational frequency of the second qubit 104. In variousembodiments, the second resonator 112 can have a second resonantfrequency. In various cases, the second resonant frequency can begreater than the first operational frequency of the first qubit 102. Invarious instances, the second resonant frequency can also be greaterthan the second operational frequency of the second qubit 104. Invarious embodiments, the first resonant frequency can be about 3 GHz(e.g., the first resonant frequency can be within any suitablemeasurement resolution and/or measurement error of 3 GHz). In variousinstances, the second resonant frequency can be about 6 GHz (e.g., thesecond resonant frequency can be within any suitable measurementresolution and/or measurement error of 6 GHz). In various aspects, theresonant frequency of a fixed-frequency microwave resonator can dependupon the shape and/or size of the fixed-frequency microwave resonator(e.g., low resonant frequencies can be obtained with long microwaveresonators, while high resonant frequencies can be obtained with shortmicrowave resonators).

In various instances, the first resonator 106, the second resonator 112,and the coupling capacitors 108, 110, 114, and 116 can be considered asa multi-resonant coupling architecture 126. As explained above, themulti-resonant coupling architecture 126 can reduce ZZ interactionbetween the first qubit 102 and the second qubit 104 withoutcorrespondingly decreasing the ZX interaction (e.g., exchange couplingJ) between the first qubit 102 and the second qubit 104. Moreover, themulti-resonant coupling architecture 126 does not require the injectionof many-pulse echoes into the system 100. Furthermore, themulti-resonant coupling architecture 126 can be constructed withouttunable-frequency elements (e.g., the first resonator 106 and the secondresonator 112 can be fixed-frequency microwave resonators). Thus, themulti-resonant coupling architecture 126 can, in various aspects, reduceZZ interaction between the first qubit 102 and the second qubit 104without correspondingly degrading coherence times of the system 100.Thus, the multi-resonant coupling architecture 126 can constitute aconcrete and tangible technical improvement over conventional systemsand/or techniques.

FIG. 2 illustrates a block diagram of an example, non-limiting system200 including one resonator that can facilitate ZZ interaction reductionin accordance with one or more embodiments described herein. As shown,in various aspects, the system 200 can comprise the first qubit 102 andthe second qubit 104, substantially as described above.

As shown, in various embodiments, the first qubit 102 can have acoupling capacitor 208. Similarly, the second qubit 104 can have acoupling capacitor 210. In various instances, the coupling capacitors208 and 210 can be any suitable coupling capacitors used in quantumcomputing systems.

In various instances, the system 200 can comprise a resonator 202. Invarious aspects, the resonator 202 can be any suitable fixed-frequencymicrowave resonator used in quantum computing systems (e.g., a busresonator). In various aspects, the resonator 202 can be any suitableλ/4 resonator. In various instances, the resonator 202 can be a long,low-frequency λ/4 resonator.

As shown, the resonator 202 can, in various embodiments, have a firstend 204 and a second end 206. In various instances, the first end 204 ofthe resonator 202 can be capacitively coupled to the first qubit 102 andcan be capacitively coupled to the second qubit 104. Specifically, invarious aspects, the first end 204 of the resonator 202 can couple tothe coupling capacitor 208 of the first qubit 102. Additionally, thefirst end 204 of the resonator 202 can also couple to the couplingcapacitor 210 of the second qubit 104. In various instances, the secondend 206 of the resonator 202 can be coupled and/or shorted to ground212.

In various embodiments, the resonator 202 can have a first harmonicfrequency. In various cases, the first harmonic frequency can be lessthan the first operational frequency of the first qubit 102. In variousinstances, the first harmonic frequency can also be less than the secondoperational frequency of the second qubit 104. In various embodiments,the resonator 202 can have a second harmonic frequency. In variouscases, the second harmonic frequency can be greater than the firstoperational frequency of the first qubit 102. In various instances, thesecond harmonic frequency can also be greater than the secondoperational frequency of the second qubit 104. In various embodiments,the first harmonic frequency can be about 2 GHz (e.g., the firstharmonic frequency can be within any suitable measurement resolutionand/or measurement error of 2 GHz). In various instances, the secondharmonic frequency can be about 6 GHz (e.g., the second harmonicfrequency can be within any suitable measurement resolution and/ormeasurement error of 6 GHz). In other words, the system 200 can have asingle resonant element (e.g., the resonator 202) and yet can have tworesonances (e.g., the first harmonic frequency and the second harmonicfrequency).

In various instances, the resonator 202, the ground 212, and thecoupling capacitors 208 and 210 can be considered as a multi-resonantcoupling architecture 214. As explained above, the multi-resonantcoupling architecture 214 can reduce ZZ interaction between the firstqubit 102 and the second qubit 104 without correspondingly decreasingthe ZX interaction (e.g., exchange coupling J) between the first qubit102 and the second qubit 104. Moreover, the multi-resonant couplingarchitecture 214 does not require the injection of many-pulse echoesinto the system 200. Furthermore, the multi-resonant couplingarchitecture 214 can be constructed without tunable-frequency elements(e.g., the resonator 202 can be a fixed-frequency microwave resonator).Thus, the multi-resonant coupling architecture 214 can, in variousaspects, reduce ZZ interaction between the first qubit 102 and thesecond qubit 104 without correspondingly degrading coherence times ofthe system 200. Thus, the multi-resonant coupling architecture 214 canconstitute a concrete and tangible technical improvement overconventional systems and/or techniques.

FIG. 3 illustrates a block diagram of an example, non-limiting system300 including a resonator and a differential direct coupler that canfacilitate ZZ interaction reduction in accordance with one or moreembodiments described herein. As shown, in various aspects, the system300 can comprise the first qubit 102 and the second qubit 104,substantially as described above.

As shown, in various embodiments, the first qubit 102 can have a firstcoupling capacitor 312 and can have a second coupling capacitor 318.Similarly, the second qubit 104 can have a first coupling capacitor 314and can have a second coupling capacitor 320. In various instances, thecoupling capacitors 312, 314, 318, and 320 can be any suitable couplingcapacitors used in quantum computing systems.

Moreover, in various instances, the first qubit 102 can have a firstpad/node 302 and can have a second pad/node 304. Similarly, the secondqubit 104 can have a first pad/node 306 and can have a second pad/node308. In various embodiments, the first pad/node 302 of the first qubit102 can be considered to be common with the second pad/node 308 of thesecond qubit 104 (e.g., common qubit pads and/or nodes). Moreover, thefirst pad/node 302 of the first qubit 102 can be considered to beopposite of the first pad/node 306 of the second qubit 104 (e.g.,opposite qubit pads and/or nodes). Similarly, in various instances, thesecond pad/node 304 of the first qubit 102 can be considered to becommon with the first pad/node 306 of the second qubit 104 (e.g., commonqubit pads and/or nodes). Additionally, the second pad/node 304 of thefirst qubit 102 can be considered to be opposite of the second pad/node308 of the second qubit 104 (e.g., opposite qubit pads and/or nodes).

In various instances, the system 300 can comprise a resonator 310 and adifferential direct coupler 316. In various aspects, the resonator 310can be any suitable fixed-frequency microwave resonator used in quantumcomputing systems (e.g., a bus resonator). In various aspects, theresonator 310 can be any suitable λ/2 resonator. In various instances,the differential direct coupler 316 can be any suitable direct couplingand/or wiring used in quantum computing systems.

As shown, the resonator 310 can, in various embodiments, capacitivelycouple the first qubit 102 to the second qubit 104. Specifically, invarious instances, the resonator 310 can have a first end (e.g., theleft-hand end of the resonator 310 as depicted in FIG. 3 ) and a secondend (e.g., the right-hand end of the resonator 310 as depicted in FIG. 3). In various cases, the first end of the resonator 310 can be coupledto the first coupling capacitor 312 of the first qubit 102. In variousaspects, the second end of the resonator 310 can be coupled to the firstcoupling capacitor 314 of the second qubit 104. Similarly, thedifferential direct coupler 316 can, in various embodiments,capacitively couple to the first qubit 102 to the second qubit 104.Specifically, in various instances, the differential direct coupler 316can have a first end (e.g., the left-hand end of the differential directcoupler 316 as depicted in FIG. 3 ) and a second end (e.g., theright-hand end of the differential direct coupler 316 as depicted inFIG. 3 ). In various cases, the first end of the differential directcoupler 316 can be coupled to the second coupling capacitor 318 of thefirst qubit 102. In various aspects, the second end of the differentialdirect coupler 316 can be coupled to the second coupling capacitor 320of the second qubit 104. As shown, in various instances, the secondcoupling capacitor 318 of the first qubit 102 can be coupled to thefirst pad/node 302 of the first qubit 102. As also shown, the secondcoupling capacitor 320 of the second qubit 104 can be coupled to thefirst pad/node 306 of the second qubit 104. Thus, in variousembodiments, the differential direct coupler 316 can be considered ascapacitively coupling together opposite pads/nodes of the first qubit102 and the second qubit 104 (e.g., the differential direct coupler 316ultimately couples the first pad/node 302 of the first qubit 102 to thefirst pad/node 306 of the second qubit 104, where the first pad/node 302of the first qubit 102 is considered to be opposite of the firstpad/node 306 of the second qubit 104).

As shown, in various instances, the resonator 310 and the differentialdirect coupler 316 can be in parallel (e.g., as opposed to in series).

In various embodiments, the resonator 310 can have a resonant frequency.In various cases, the resonant frequency can be greater than the firstoperational frequency of the first qubit 102. In various instances, theresonant frequency can also be greater than the second operationalfrequency of the second qubit 104. In various embodiments, the resonantfrequency can be about 6 GHz (e.g., the resonant frequency can be withinany suitable measurement resolution and/or measurement error of 6 GHz).In various aspects, the differential direct coupler 316 can be anysuitable, short section of transmission line (e.g., short in the sensethat its resonance frequency is greater than about 30 GHz), which cangive rise to a frequency-independent coupling at typical transmon qubitfrequencies of around 5 GHz.

In various instances, the resonator 310, the differential direct coupler316, and the coupling capacitors 312, 314, 318, and 320 can beconsidered as a multi-resonant coupling architecture 322. As explainedabove, the multi-resonant coupling architecture 322 can reduce ZZinteraction between the first qubit 102 and the second qubit 104 withoutcorrespondingly decreasing the ZX interaction (e.g., exchange couplingJ) between the first qubit 102 and the second qubit 104. Moreover, themulti-resonant coupling architecture 322 does not require the injectionof many-pulse echoes into the system 300. Furthermore, themulti-resonant coupling architecture 322 can be constructed withouttunable-frequency elements (e.g., the resonator 310 can be afixed-frequency microwave resonator, and the differential direct coupler316 can be any suitable, short section of transmission line (e.g., shortin the sense that its resonance frequency is greater than about 30 GHz),which can give rise to a frequency-independent coupling at typicaltransmon qubit frequencies of around 5 GHz, or can be considered as anon-resonant structure). Thus, the multi-resonant coupling architecture322 can, in various aspects, reduce ZZ interaction between the firstqubit 102 and the second qubit 104 without correspondingly degradingcoherence times of the system 300. Moreover, in various embodiments, thedifferential direct coupler 316 can be a short and/or compact directcoupler, and the resonator 310 can be a short microwave resonator (e.g.,a microwave resonator that can generate a high resonant frequency like 6GHz can be shorter and/or more compact than a microwave resonator thatcan generate a low resonant frequency like 3 GHz). Thus, in variousinstances, the multi-resonant coupling architecture 322 can be verycompact (and thus amenable to scaling to large device sizes), ascompared to conventional systems and/or techniques. Thus, themulti-resonant coupling architecture 322 can constitute a concrete andtangible technical improvement over conventional systems and/ortechniques.

FIG. 4 illustrates a block diagram of an example, non-limiting system400 including a resonator and a direct coupler that can facilitate ZZinteraction reduction in accordance with one or more embodimentsdescribed herein. As shown, in various aspects, the system 400 cancomprise the first qubit 102 and the second qubit 104, substantially asdescribed above.

As shown, in various embodiments, the first qubit 102 can have a firstcoupling capacitor 408 and can have a second coupling capacitor 416.Similarly, the second qubit 104 can have a first coupling capacitor 410and can have a second coupling capacitor 418. In various instances, thecoupling capacitors 408, 410, 416, and 418 can be any suitable couplingcapacitors used in quantum computing systems.

Moreover, in various instances, the first qubit 102 can have the firstpad/node 302 and can have a second pad/node 304, substantially asdescribed above. Similarly, the second qubit 104 can have a firstpad/node 306 and can have a second pad/node 308, substantially asdescribed above. As explained above, in various embodiments, the firstpad/node 302 of the first qubit 102 can be considered to be common withthe second pad/node 308 of the second qubit 104 (e.g., common qubit padsand/or nodes). Moreover, the first pad/node 302 of the first qubit 102can be considered to be opposite of the first pad/node 306 of the secondqubit 104 (e.g., opposite qubit pads and/or nodes). Similarly, invarious instances, the second pad/node 304 of the first qubit 102 can beconsidered to be common with the first pad/node 306 of the second qubit104 (e.g., common qubit pads and/or nodes). Additionally, the secondpad/node 304 of the first qubit 102 can be considered to be opposite ofthe second pad/node 308 of the second qubit 104 (e.g., opposite qubitpads and/or nodes).

In various instances, the system 400 can comprise a resonator 402 and adirect coupler 414. In various aspects, the resonator 402 can be anysuitable fixed-frequency microwave resonator used in quantum computingsystems (e.g., a bus resonator). In various aspects, the resonator 402can be any suitable λ/4 resonator. In various instances, the directcoupler 414 can be any suitable direct coupling and/or wiring used inquantum computing systems.

As shown, the resonator 402 can, in various embodiments, have a firstend 404 and can have a second end 406. In various cases, the first end404 of the resonator 402 can be capacitively coupled to the first qubit102 and can be capacitively coupled to the second qubit 104.Specifically, in various instances, the first end 404 of the resonator402 can be coupled to the first coupling capacitor 408 of the firstqubit 102. Additionally, in various aspects, the first end 404 of theresonator 402 can also be coupled to the first coupling capacitor 410 ofthe second qubit 104. In various instances, the second end 406 of theresonator 402 can be coupled and/or shorted to ground 412.

In various embodiments, the direct coupler 414 can capacitively couplethe first qubit 102 to the second qubit 104. Specifically, in variousinstances, the direct coupler 414 can have a first end (e.g., theleft-hand end of the direct coupler 414 as depicted in FIG. 4 ) and asecond end (e.g., the right-hand end of the direct coupler 414 asdepicted in FIG. 4 ). In various cases, the first end of the directcoupler 414 can be coupled to the second coupling capacitor 416 of thefirst qubit 102. In various aspects, the second end of the directcoupler 414 can be coupled to the second coupling capacitor 418 of thesecond qubit 104. As shown, in various instances, the second couplingcapacitor 416 of the first qubit 102 can be coupled to the secondpad/node 304 of the first qubit 102. As also shown, the second couplingcapacitor 418 of the second qubit 104 can be coupled to the firstpad/node 306 of the second qubit 104. Thus, in various embodiments, thedirect coupler 414 can be considered as capacitively coupling togethercommon pads/nodes of the first qubit 102 and the second qubit 104 (e.g.,the direct coupler 414 ultimately couples the second pad/node 304 of thefirst qubit 102 to the first pad/node 306 of the second qubit 104, wherethe second pad/node 304 of the first qubit 102 is considered to becommon with the first pad/node 306 of the second qubit 104).

In various embodiments, the resonator 402 can have a resonant frequency.In various cases, the resonant frequency can be greater than the firstoperational frequency of the first qubit 102. In various instances, theresonant frequency can also be greater than the second operationalfrequency of the second qubit 104. In various embodiments, the resonantfrequency can be about 6 GHz (e.g., the resonant frequency can be withinany suitable measurement resolution and/or measurement error of 6 GHz).In various aspects, the direct coupler 414 can be any suitable, shortsection of transmission line (e.g., short in the sense that itsresonance frequency is greater than about 30 GHz), which can give riseto a frequency-independent coupling at typical transmon qubitfrequencies of around 5 GHz.

In various instances, the resonator 402, the direct coupler 414, theground 412, and the coupling capacitors 408, 410, 416, and 418 can beconsidered as a multi-resonant coupling architecture 420. As explainedabove, the multi-resonant coupling architecture 420 can reduce ZZinteraction between the first qubit 102 and the second qubit 104 withoutcorrespondingly decreasing the ZX interaction (e.g., exchange couplingJ) between the first qubit 102 and the second qubit 104. Moreover, themulti-resonant coupling architecture 420 does not require the injectionof many-pulse echoes into the system 400. Furthermore, themulti-resonant coupling architecture 420 can be constructed withouttunable-frequency elements (e.g., the resonator 402 can be afixed-frequency microwave resonator, and the direct coupler 414 can beany suitable, short section of transmission line (e.g., short in thesense that its resonance frequency is greater than about 30 GHz), whichcan give rise to a frequency-independent coupling at typical transmonqubit frequencies of around 5 GHz, or can be considered as anon-resonant structure). Thus, the multi-resonant coupling architecture420 can, in various aspects, reduce ZZ interaction between the firstqubit 102 and the second qubit 104 without correspondingly degradingcoherence times of the system 400. Moreover, in various embodiments, thedirect coupler 414 can be a short and/or compact direct coupler, and theresonator 402 can be a short microwave resonator (e.g., a microwaveresonator that can have a high resonator frequency like 6 GHz can beshorter and/or more compact than a microwave resonator that can have alow resonant frequency like 2 GHz). Thus, in various instances, themulti-resonant coupling architecture 420 can be very compact (and thusamenable to scaling to large device sizes), as compared to conventionalsystems and/or techniques. Thus, the multi-resonant couplingarchitecture 420 can constitute a concrete and tangible technicalimprovement over conventional systems and/or techniques.

FIGS. 5-6 illustrate example, non-limiting graphs 500 and 600 thatdepict the ZZ interaction reduction facilitated by one or moreembodiments described herein.

As shown in FIGS. 5-6 , the graph 500 depicts computational simulationresults of various embodiments of the invention, and the graph 600depicts computational simulation results of various embodiments of theinvention as compared to computational simulation results ofconventional qubit coupling techniques. The inventors of variousembodiments of the invention ran these computational simulations tocalculate and/or approximate the ZX interaction (e.g., coupling strengthand/or exchange coupling J) between two fixed-frequency superconductingqubits and to calculate and/or approximate the ZZ interaction betweenthe two fixed-frequency superconducting qubits. In some of thesimulations, the inventors assumed that the two fixed-frequencysuperconducting qubits were conventionally coupled. In other of thesimulations, the inventors assumed that the two fixed-frequencysuperconducting qubits were coupled via a multi-resonant couplingarchitecture such as the multi-resonant coupling architecture 420. Forthese simulations, the inventors used the following values: g=80 MHzwherein g denotes the coupling strength between the transmon qubit andthe bus resonator, f₀=5000 MHz where f₀ denotes the frequency of theupper transmon qubit, α=−320 MHz where a denotes the transmon qubitanharmonicity, ω=6100 MHz where ω denotes the bus resonator frequency,and J₀=4.5 MHz where J₀ denotes the transmon direct coupler exchangeinteraction. Moreover, the frequency separation between the twofixed-frequency superconducting qubits was set at 150 MHz. In variousaspects, the simulations were run with a drive signal set at 60 MHz.

The graph 500 depicts a subset of the simulation results for variousembodiments of the multi-resonant coupling architecture 420. As shown,the graph 500 includes a line 502 indicating and/or corresponding to thevalue of the ZX interaction between the two qubits as a function of theupper qubit operational frequency when the qubits are coupled by themulti-resonant coupling architecture 420. That is, the line 502corresponds to the variable “ZX (60 MHz Drive)” as noted in the legendof FIG. 5 . Also as shown, the graph 500 includes a line 504 indicatingand/or corresponding to the value of the ZZ interaction between the twoqubits as a function of the upper qubit operational frequency when thequbits are coupled by the multi-resonant coupling architecture 420. Thatis, the line 504 corresponds to the variable “ZZ” as noted in the legendof FIG. 5 . As indicated by numeral 506, there is a particular range ofupper qubit operational frequencies (e.g., between 5150 MHz and 5200MHz) where the ZZ interaction significantly drops (e.g., a zero ZZinteraction) and where the ZX interaction does not significantly drop(e.g., a non-zero ZX interaction). In other words, the graph 500illustrates a particular frequency band in which various embodiments ofthe invention cause a significant reduction in ZZ interaction withoutcausing a corresponding reduction in ZX interaction. This drop in the ZZinteraction without a corresponding drop in the ZX interaction isfacilitated by various embodiments of the invention. Moreover, sincevarious embodiments of the invention do not require echoing and/ortunable-frequency elements, various embodiments of the invention canfacilitate such ZZ reduction without the corresponding coherencedegradation that normally accompanies conventional systems and/ortechniques.

The graph 600 is similar to the graph 500, in that the graph 600 depictsthe simulation results for various embodiments of the multi-resonantcoupling architecture 420. However, the graph 600 depicts such resultsover a larger frequency band (e.g., from an upper qubit operationalfrequency of 4 GHz to 5.75 GHz) and also includes the results associatedwith conventional coupling techniques. As shown, the graph 600 includesa line 602 that indicates and/or corresponds to the value of the ZXinteraction between the two qubits as a function of the upper qubitoperational frequency when the qubits are conventionally coupled. Thatis, the line 602 corresponds to the variable “J (impedance)” as noted inthe legend of FIG. 6 . The graph 600 also includes a line 604 thatindicates and/or corresponds to the value of the ZZ interaction betweenthe two qubits as a function of the upper qubit operation frequency whenthe two qubits are conventionally coupled. That is, the line 604corresponds to the variable “ZZ (impedance)” as noted in the legend ofFIG. 6 . As shown, conventional coupling techniques can cause both theZX interaction and the ZZ interaction to significantly decrease in aparticular frequency band (e.g., between 4.5 GHz and 4.75 GHz). However,as shown, there is no frequency band in which the conventional couplingtechniques cause the ZZ interaction to significantly drop without acorresponding drop in the ZX interaction. As shown, conventionalcoupling techniques yield a low ZZ interaction only in operating pointswhere the ZX interaction is also weak. However, as shown, variousembodiments of the invention can yield operating points where the ZZinteraction is weak despite a nonzero and/or non-weak ZX interaction.

As shown, the graph 600 includes a line 606 that indicates and/orcorresponds to the value of the ZX interaction between the two qubits asa function of the upper qubit operational frequency when the qubits arecoupled by the multi-resonant coupling architecture 420. That is, theline 606 corresponds to the variable “J (gs)” as noted in the legend ofFIG. 6 . Additionally, the graph 600 includes a line 608 that indicatesand/or corresponds to the value of the ZZ interaction between the twoqubits as a function of the upper qubit operational frequency when thequbits are coupled by the multi-resonant coupling architecture 420. Thatis, the line 608 corresponds to the variable “ZZ (gs)” as noted in thelegend of FIG. 6 . As shown, various embodiments of the multi-resonantcoupling architecture 420 can cause both the ZX interaction and the ZZinteraction to significantly decrease in a particular frequency band(e.g., between 4.5 GHz and 4.75 GHz). Also as shown, various embodimentsof the multi-resonant coupling architecture 420 can cause the ZZinteraction to significantly decrease without a corresponding decreasein the ZX interaction in a different frequency band (e.g., between 5 GHzand 5.25 GHz).

These results can be compared to demonstrate the benefits of variousembodiments of the invention over conventional systems and/ortechniques. As shown, the line 606 is nearly identical to the line 602.In other words, the multi-resonant coupling architecture 420 can providea nearly identical ZX interaction (e.g., a coupling strength and/orexchange coupling J) as conventional coupling techniques. However, asshown, the line 608 is significantly lower than the line 604 for a broadfrequency band (e.g., from about 4.8 GHz to 5.75 GHz). Indeed, as shownin the graph 600, the line 608 is nearly a full order of magnitude belowthe line 604 from about 5 GHz to 5.75 GHz, and the line 608 is nearlytwo orders of magnitude below the line 604 in a narrow frequency bandbetween 5 GHz and 5.25 GHz, as shown. Such improved performance clearlyshows that various embodiments of the invention constitute concrete andtangible technical improvements over the prior art.

Note that the graphs 500 and 600 are exemplary and non-limiting. Invarious aspects, the zero ZZ interaction and the non-zero ZX interactioncan occur in different upper qubit operational frequency than thosedepicted in FIGS. 5-6 , based on various parameters corresponding to themulti-resonant coupling architecture that is used to couple the qubitsand/or based on the operating environment of the coupled qubits (e.g.,different resonant frequencies of the λ/2 and/or λ/4 couplers, differentdrive signals). Also, note that the graphs 500 and 600 show theparticular simulation results for various embodiments of themulti-resonant coupling architecture 420. However, very similarsimulation results were obtained by the inventors for the various otherembodiments of the invention (e.g., for the multi-resonant couplingarchitectures 126, 214, and 322). Since the results were nearlyidentical, the other simulation results are omitted for sake of brevity.

FIG. 7 illustrates a block diagram of an example, non-limiting qubitarray 700 that can facilitate ZZ interaction reduction in accordancewith one or more embodiments described herein.

As shown in FIG. 7 , various embodiments of the invention can beimplemented to create a qubit array 700 (e.g., a two-dimensional arrayof coupled qubits). As shown, the qubit array 700 can, in variousaspects, comprise the qubits Q1 to Q4. In various aspects, the qubits Q1to Q4 can be any suitable types and/or combinations of types ofsuperconducting qubits (e.g., the qubits Q1 to Q4 can be the same typeof qubit, and/or can be different types of qubits). In variousinstances, the qubit array 700 can be a square and/or lattice array(e.g., two rows of two qubits). In various aspects, the qubit array 700can be arranged in any other suitable configuration and/or shape (e.g.,rectangle, triangle, circle). Although FIG. 7 depicts only four qubits(e.g., Q1 to Q4) in the qubit array 700, this is for illustration only.In various instances, any suitable number of qubits can be implementedin the qubit array 700. In various aspects, the qubits Q1 to Q4 can bearranged in the qubit array 700 on any suitable quantum computingsubstrate (not shown).

As shown, in various embodiments, any qubit in the qubit array 700 canbe coupled to some and/or all of its nearest-neighbor qubits (and/orsome and/or all of its next-nearest neighbor qubits, in some cases) byany suitable multi-resonant coupling architecture as described herein.For example, as shown, the qubit Q1 can be coupled to the qubit Q2 via amulti-resonant coupling architecture 126, as explained in relation toFIG. 1 . As shown, the qubit Q1 can also be coupled to the qubit Q3 by amulti-resonant coupling architecture 322, as explained in relation toFIG. 3 (e.g., for the sake of illustrative simplicity, FIG. 7 does notillustrate the differential nature of the multi-resonant couplingarchitecture 322; however, such differential nature is amply depictedand described in relation to FIG. 3 ). As shown, the qubit Q2 can alsobe coupled to the qubit Q4 by a multi-resonant coupling architecture420, as explained in relation to FIG. 4 . As shown, the qubit Q3 can becoupled to the qubit Q4 by a multi-resonant coupling architecture 214,as explained in relation to FIG. 2 . Although not illustrated in FIG. 7, one or more conventional couplers can also be implemented in the qubitarray 700, in various instances.

In various aspects, FIG. 7 depicts a non-limiting example of how one ormore of the multi-resonant coupling architectures depicted in FIGS. 1-4(e.g., multi-resonant coupling architectures, 126, 214, 322, and 420)can be implemented to create two-dimensional arrays of coupled qubitswith reduced ZZ interactions.

FIG. 8 illustrates a flow diagram of an example, non-limiting method 800including two resonators that can facilitate ZZ interaction reduction inaccordance with one or more embodiments described herein.

In various embodiments, act 802 can include capacitively coupling afirst qubit (e.g., 102) to a second qubit (e.g., 104) via a firstresonator (e.g., 106). In various instances, the first qubit can have afirst operational frequency, and the second qubit can have a secondoperational frequency. In various aspects, the first resonator can havea first resonant frequency that is less than the first operationalfrequency and that is less than the second operational frequency.

In various instances, act 804 can include capacitively coupling thefirst qubit to the second qubit via a second resonator (e.g., 112) thatis in parallel with the first resonator. In various instances, thesecond resonator can have a second resonant frequency that is greaterthan the first operational frequency and that is greater than the secondoperational frequency. In various aspects, the first resonator and thesecond resonator can be λ/2 resonators. In various instances, the firstresonant frequency, the second resonant frequency, the first operationalfrequency, and the second operational frequency can be fixed.

FIG. 9 illustrates a flow diagram of an example, non-limiting method 900including one resonator that can facilitate ZZ interaction reduction inaccordance with one or more embodiments described herein.

In various embodiments, act 902 can include capacitively coupling afirst end (e.g., 204) of a resonator (e.g., 202) to a first qubit (e.g.,102) and to a second qubit (e.g., 104). In various instances, the firstqubit can have a first operational frequency, and the second qubit canhave a second operational frequency.

In various instances, act 904 can include coupling a second end (e.g.,206) of the resonator to ground (e.g., 212). In various instances, theresonator can have a first harmonic frequency that is less than thefirst operational frequency and that is less than the second operationalfrequency. In various aspects, the resonator can have a second harmonicfrequency that is greater than the first operational frequency and thatis greater than the second operational frequency. In various instances,the resonator can be a λ/4 resonator. In various instances, the firstharmonic frequency, the second harmonic frequency, the first operationalfrequency, and the second operational frequency can be fixed.

FIG. 10 illustrates a flow diagram of an example, non-limiting method1000 including a resonator and a differential direct coupler that canfacilitate ZZ interaction reduction in accordance with one or moreembodiments described herein.

In various embodiments, act 1002 can include capacitively coupling afirst qubit (e.g., 102) to a second qubit (e.g., 104) via a resonator(e.g., 310). In various instances, the first qubit can have a firstoperational frequency, the second qubit can have a second operationalfrequency, and the resonator can have a resonant frequency that isgreater than the first operational frequency and that is greater thanthe second operational frequency.

In various instances, act 1004 can include capacitively coupling thefirst qubit to the second qubit via a differential direct coupler (e.g.,316) that is in parallel with the resonator. In various instances, thedifferential direct coupler can capacitively couple opposite pads (e.g.,302 and 306) of the first qubit and the second qubit. In various cases,the resonator can be a λ/2 resonator. In various instances, the resonantfrequency, the first operational frequency, and the second operationalfrequency can be fixed.

FIG. 11 illustrates a flow diagram of an example, non-limiting method1100 including a resonator and a direct coupler that can facilitate ZZinteraction reduction in accordance with one or more embodimentsdescribed herein.

In various embodiments, act 1102 can include capacitively coupling afirst qubit (e.g., 102) to a second qubit (e.g., 104) via a resonator(e.g., 402). In various cases, a first end (e.g., 404) of the resonatorcan be capacitively coupled to the first qubit and to the second qubit,and a second end (e.g., 406) of the resonator can be coupled to ground(e.g., 412). In various cases, the first qubit can have a firstoperational frequency, the second qubit can have a second operationalfrequency, and the resonator can have a resonant frequency that isgreater than the first operational frequency and that is greater thanthe second operational frequency.

In various instances, act 1104 can include capacitively coupling thefirst qubit to the second qubit via a direct coupler (e.g., 414). Invarious instances, the direct coupler can capacitively couple commonpads (e.g., 304 and 306) of the first qubit and the second qubit. Invarious cases, the resonator can be a λ/4 resonator. In variousinstances, the resonant frequency, the first operational frequency, andthe second operational frequency can be fixed.

FIG. 12 illustrates a flow diagram of an example, non-limiting method1200 that can facilitate ZZ interaction reduction in accordance with oneor more embodiments described herein.

In various embodiments, act 1202 can include capacitively coupling afirst qubit (e.g., 102) to a second qubit (e.g., 104) via a non-tunablemulti-resonant architecture (e.g., a coupling architecture as shown inFIGS. 1-4 ). In various instances, the multi-resonant architecture caninclude a first pole that is greater than both a first operationalfrequency of the first qubit and a second operational frequency of thesecond qubit (e.g., in FIG. 1 , the second resonant frequency of thesecond resonator 112 can be the first pole; in FIG. 2 , the secondharmonic frequency of the resonator 202 can be the first pole; in FIG. 3, the resonant frequency of the resonator 310 can be the first pole; inFIG. 4 , the resonant frequency of the resonator 402 can be the firstpole). In various aspects, the multi-resonant architecture can include asecond pole that is less than both the first operational frequency andthe second operational frequency (e.g., in FIG. 1 , the first resonantfrequency of the first resonator 106 can be the second pole; in FIG. 2 ,the first harmonic frequency of the resonator 202 can be the secondpole). In various other aspects, rather than the second pole, themulti-resonant architecture can instead include a direct coupling term(e.g., the differential direct coupler 316 in FIG. 3 , or the directcoupler 414 in FIG. 4 ). In various cases, the multi-resonantarchitecture can exhibit a zero coupling strength and a zero ZZinteraction in a first set of qubit frequencies (e.g., between 4.5 GHzand 4.75 GHz as shown in FIG. 6 ). In various instances, themulti-resonant architecture can exhibit a non-zero coupling strength anda zero ZZ interaction in a second set of qubit frequencies (e.g.,between 5 GHz and 5.25 GHz as shown in FIG. 6 ).

Various embodiments of the invention can reduce unwanted ZZ interactionswhile preserving desirable ZX interactions. In various instances, thiscan be accomplished by a multi-resonant coupling architecture that hastwo fixed-frequency elements. In various aspects, the detuning betweenthe two fixed-frequency elements and the qubits can be different, whichcan facilitate the suppression of the ZZ interaction. In various otherinstances, this can be accomplished by a multi-element coupler thatincludes a resonator and a short capacitive coupler. In variousinstances, the two qubits' interactions with the elements can bedifferent, which can, in certain frequency bands, result in cancellationof the undesired ZZ interaction.

In various instances, the following sample experiment can be conducted.Two qubits can be coupled together via any embodiment of the invention(e.g., via any multi-resonant coupling architecture discussed herein).The qubits can be weakly-tunable so that the parameters and/orperformance of the multi-resonant coupling architecture can beinvestigated. For various combinations of pairs of qubit frequencies,the exchange coupling J and the ZZ interaction can be tested and/orrecorded (e.g., J can be estimated from the ZX rate of cross-resonance;ZZ interaction can be measured by a Pi-Ramsey experiment). This canallow the ZZ cancellation points to be mapped out for a given coupler.Then, ac-Stark shifts can be employed to tune the weakly-tunable qubitsinto a desired regime. Finally, a cross-resonance gate can be operatedwith the qubits in the ZZ cancellation bandwidth.

In some cases, a multi-resonant coupling architecture can include twoλ/2 resonators (e.g., as shown in FIG. 1 ), with one having a resonantfrequency of 4 GHz and the other having a resonant frequency of 6 GHz.

In some cases, a multi-resonant coupling architecture can include asingle λ/4 resonator (e.g., as shown in FIG. 2 ), having a firstharmonic at 2 GHz and a second harmonic at 6 GHz, which can combine toreduce and/or suppress ZZ interactions.

In some cases, a multi-resonant coupling architecture can include a λ/2resonator at 6 GHz and a direct capacitive connection betweendifferential pads of the qubits (e.g., as shown in FIG. 3 ). In variousinstances, the couplings via the two paths can be balanced so thatexchange coupling J approaches zero around an upper qubit operationalfrequency of 4.7 GHz, which can cause a zero ZZ interaction and anon-zero exchange coupling J at about 5 GHz.

In some cases, a multi-resonant coupling architecture can include a λ/4resonator at 6 GHz and a direct capacitive connection between commonpads of the qubits (e.g., as shown in FIG. 4 ).

Various embodiments of the invention can provide for a multi-resonantcoupling architecture that can include one or more coupling elementswhose frequency response results in a cancellation of state-dependentcoupling at the qubit frequencies while maintaining a finitestate-independent coupling.

In order to provide additional context for various embodiments describedherein, FIG. 13 and the following discussion are intended to provide ageneral description of a suitable computing environment 1300 in whichthe various embodiments of the embodiment described herein can beimplemented. While the embodiments have been described above in thegeneral context of computer-executable instructions that can run on oneor more computers, those skilled in the art will recognize that theembodiments can be also implemented in combination with other programmodules and/or as a combination of hardware and software.

Generally, program modules include routines, programs, components, datastructures, etc., that perform particular tasks or implement particularabstract data types. Moreover, those skilled in the art will appreciatethat the inventive methods can be practiced with other computer systemconfigurations, including single-processor or multiprocessor computersystems, minicomputers, mainframe computers, Internet of Things (IoT)devices, distributed computing systems, as well as personal computers,hand-held computing devices, microprocessor-based or programmableconsumer electronics, and the like, each of which can be operativelycoupled to one or more associated devices.

The illustrated embodiments of the embodiments herein can be alsopracticed in distributed computing environments where certain tasks areperformed by remote processing devices that are linked through acommunications network. In a distributed computing environment, programmodules can be located in both local and remote memory storage devices.

Computing devices typically include a variety of media, which caninclude computer-readable storage media, machine-readable storage media,and/or communications media, which two terms are used herein differentlyfrom one another as follows. Computer-readable storage media ormachine-readable storage media can be any available storage media thatcan be accessed by the computer and includes both volatile andnonvolatile media, removable and non-removable media. By way of example,and not limitation, computer-readable storage media or machine-readablestorage media can be implemented in connection with any method ortechnology for storage of information such as computer-readable ormachine-readable instructions, program modules, structured data orunstructured data.

Computer-readable storage media can include, but are not limited to,random access memory (RAM), read only memory (ROM), electricallyerasable programmable read only memory (EEPROM), flash memory or othermemory technology, compact disk read only memory (CD-ROM), digitalversatile disk (DVD), Blu-ray disc (BD) or other optical disk storage,magnetic cassettes, magnetic tape, magnetic disk storage or othermagnetic storage devices, solid state drives or other solid statestorage devices, or other tangible and/or non-transitory media which canbe used to store desired information. In this regard, the terms“tangible” or “non-transitory” herein as applied to storage, memory orcomputer-readable media, are to be understood to exclude onlypropagating transitory signals per se as modifiers and do not relinquishrights to all standard storage, memory or computer-readable media thatare not only propagating transitory signals per se.

Computer-readable storage media can be accessed by one or more local orremote computing devices, e.g., via access requests, queries or otherdata retrieval protocols, for a variety of operations with respect tothe information stored by the medium.

Communications media typically embody computer-readable instructions,data structures, program modules or other structured or unstructureddata in a data signal such as a modulated data signal, e.g., a carrierwave or other transport mechanism, and includes any information deliveryor transport media. The term “modulated data signal” or signals refersto a signal that has one or more of its characteristics set or changedin such a manner as to encode information in one or more signals. By wayof example, and not limitation, communication media include wired media,such as a wired network or direct-wired connection, and wireless mediasuch as acoustic, RF, infrared and other wireless media.

With reference again to FIG. 13 , the example environment 1300 forimplementing various embodiments of the aspects described hereinincludes a computer 1302, the computer 1302 including a processing unit1304, a system memory 1306 and a system bus 1308. The system bus 1308couples system components including, but not limited to, the systemmemory 1306 to the processing unit 1304. The processing unit 1304 can beany of various commercially available processors. Dual microprocessorsand other multi-processor architectures can also be employed as theprocessing unit 1304.

The system bus 1308 can be any of several types of bus structure thatcan further interconnect to a memory bus (with or without a memorycontroller), a peripheral bus, and a local bus using any of a variety ofcommercially available bus architectures. The system memory 1306includes ROM 1310 and RAM 1312. A basic input/output system (BIOS) canbe stored in a non-volatile memory such as ROM, erasable programmableread only memory (EPROM), EEPROM, which BIOS contains the basic routinesthat help to transfer information between elements within the computer1302, such as during startup. The RAM 1312 can also include a high-speedRAM such as static RAM for caching data.

The computer 1302 further includes an internal hard disk drive (HDD)1314 (e.g., EIDE, SATA), one or more external storage devices 1316(e.g., a magnetic floppy disk drive (FDD) 1316, a memory stick or flashdrive reader, a memory card reader, etc.) and a drive 1320, e.g., suchas a solid state drive, an optical disk drive, which can read or writefrom a disk 1322, such as a CD-ROM disc, a DVD, a BD, etc.Alternatively, where a solid state drive is involved, disk 1322 wouldnot be included, unless separate. While the internal HDD 1314 isillustrated as located within the computer 1302, the internal HDD 1314can also be configured for external use in a suitable chassis (notshown). Additionally, while not shown in environment 1300, a solid statedrive (SSD) could be used in addition to, or in place of, an HDD 1314.The HDD 1314, external storage device(s) 1316 and drive 1320 can beconnected to the system bus 1308 by an HDD interface 1324, an externalstorage interface 1326 and a drive interface 1328, respectively. Theinterface 1324 for external drive implementations can include at leastone or both of Universal Serial Bus (USB) and Institute of Electricaland Electronics Engineers (IEEE) 1394 interface technologies. Otherexternal drive connection technologies are within contemplation of theembodiments described herein.

The drives and their associated computer-readable storage media providenonvolatile storage of data, data structures, computer-executableinstructions, and so forth. For the computer 1302, the drives andstorage media accommodate the storage of any data in a suitable digitalformat. Although the description of computer-readable storage mediaabove refers to respective types of storage devices, it should beappreciated by those skilled in the art that other types of storagemedia which are readable by a computer, whether presently existing ordeveloped in the future, could also be used in the example operatingenvironment, and further, that any such storage media can containcomputer-executable instructions for performing the methods describedherein.

A number of program modules can be stored in the drives and RAM 1312,including an operating system 1330, one or more application programs1332, other program modules 1334 and program data 1336. All or portionsof the operating system, applications, modules, and/or data can also becached in the RAM 1312. The systems and methods described herein can beimplemented utilizing various commercially available operating systemsor combinations of operating systems.

Computer 1302 can optionally comprise emulation technologies. Forexample, a hypervisor (not shown) or other intermediary can emulate ahardware environment for operating system 1330, and the emulatedhardware can optionally be different from the hardware illustrated inFIG. 13 . In such an embodiment, operating system 1330 can comprise onevirtual machine (VM) of multiple VMs hosted at computer 1302.Furthermore, operating system 1330 can provide runtime environments,such as the Java runtime environment or the .NET framework, forapplications 1332. Runtime environments are consistent executionenvironments that allow applications 1332 to run on any operating systemthat includes the runtime environment. Similarly, operating system 1330can support containers, and applications 1332 can be in the form ofcontainers, which are lightweight, standalone, executable packages ofsoftware that include, e.g., code, runtime, system tools, systemlibraries and settings for an application.

Further, computer 1302 can be enable with a security module, such as atrusted processing module (TPM). For instance with a TPM, bootcomponents hash next in time boot components, and wait for a match ofresults to secured values, before loading a next boot component. Thisprocess can take place at any layer in the code execution stack ofcomputer 1302, e.g., applied at the application execution level or atthe operating system (OS) kernel level, thereby enabling security at anylevel of code execution.

A user can enter commands and information into the computer 1302 throughone or more wired/wireless input devices, e.g., a keyboard 1338, a touchscreen 1340, and a pointing device, such as a mouse 1342. Other inputdevices (not shown) can include a microphone, an infrared (IR) remotecontrol, a radio frequency (RF) remote control, or other remote control,a joystick, a virtual reality controller and/or virtual reality headset,a game pad, a stylus pen, an image input device, e.g., camera(s), agesture sensor input device, a vision movement sensor input device, anemotion or facial detection device, a biometric input device, e.g.,fingerprint or iris scanner, or the like. These and other input devicesare often connected to the processing unit 1304 through an input deviceinterface 1344 that can be coupled to the system bus 1308, but can beconnected by other interfaces, such as a parallel port, an IEEE 1394serial port, a game port, a USB port, an IR interface, a BLUETOOTH®interface, etc.

A monitor 1346 or other type of display device can be also connected tothe system bus 1308 via an interface, such as a video adapter 1348. Inaddition to the monitor 1346, a computer typically includes otherperipheral output devices (not shown), such as speakers, printers, etc.

The computer 1302 can operate in a networked environment using logicalconnections via wired and/or wireless communications to one or moreremote computers, such as a remote computer(s) 1350. The remotecomputer(s) 1350 can be a workstation, a server computer, a router, apersonal computer, portable computer, microprocessor-based entertainmentappliance, a peer device or other common network node, and typicallyincludes many or all of the elements described relative to the computer1302, although, for purposes of brevity, only a memory/storage device1352 is illustrated. The logical connections depicted includewired/wireless connectivity to a local area network (LAN) 1354 and/orlarger networks, e.g., a wide area network (WAN) 1356. Such LAN and WANnetworking environments are commonplace in offices and companies, andfacilitate enterprise-wide computer networks, such as intranets, all ofwhich can connect to a global communications network, e.g., theInternet.

When used in a LAN networking environment, the computer 1302 can beconnected to the local network 1354 through a wired and/or wirelesscommunication network interface or adapter 1358. The adapter 1358 canfacilitate wired or wireless communication to the LAN 1354, which canalso include a wireless access point (AP) disposed thereon forcommunicating with the adapter 1358 in a wireless mode.

When used in a WAN networking environment, the computer 1302 can includea modem 1360 or can be connected to a communications server on the WAN1356 via other means for establishing communications over the WAN 1356,such as by way of the Internet. The modem 1360, which can be internal orexternal and a wired or wireless device, can be connected to the systembus 1308 via the input device interface 1344. In a networkedenvironment, program modules depicted relative to the computer 1302 orportions thereof, can be stored in the remote memory/storage device1352. It will be appreciated that the network connections shown areexample and other means of establishing a communications link betweenthe computers can be used.

When used in either a LAN or WAN networking environment, the computer1302 can access cloud storage systems or other network-based storagesystems in addition to, or in place of, external storage devices 1316 asdescribed above, such as but not limited to a network virtual machineproviding one or more aspects of storage or processing of information.Generally, a connection between the computer 1302 and a cloud storagesystem can be established over a LAN 1354 or WAN 1356 e.g., by theadapter 1358 or modem 1360, respectively. Upon connecting the computer1302 to an associated cloud storage system, the external storageinterface 1326 can, with the aid of the adapter 1358 and/or modem 1360,manage storage provided by the cloud storage system as it would othertypes of external storage. For instance, the external storage interface1326 can be configured to provide access to cloud storage sources as ifthose sources were physically connected to the computer 1302.

The computer 1302 can be operable to communicate with any wirelessdevices or entities operatively disposed in wireless communication,e.g., a printer, scanner, desktop and/or portable computer, portabledata assistant, communications satellite, any piece of equipment orlocation associated with a wirelessly detectable tag (e.g., a kiosk,news stand, store shelf, etc.), and telephone. This can include WirelessFidelity (Wi-Fi) and BLUETOOTH® wireless technologies. Thus, thecommunication can be a predefined structure as with a conventionalnetwork or simply an ad hoc communication between at least two devices.

The present invention may be a system, a method, an apparatus and/or acomputer program product at any possible technical detail level ofintegration. The computer program product can include a computerreadable storage medium (or media) having computer readable programinstructions thereon for causing a processor to carry out aspects of thepresent invention. The computer readable storage medium can be atangible device that can retain and store instructions for use by aninstruction execution device. The computer readable storage medium canbe, for example, but is not limited to, an electronic storage device, amagnetic storage device, an optical storage device, an electromagneticstorage device, a semiconductor storage device, or any suitablecombination of the foregoing. A non-exhaustive list of more specificexamples of the computer readable storage medium can also include thefollowing: a portable computer diskette, a hard disk, a random accessmemory (RAM), a read-only memory (ROM), an erasable programmableread-only memory (EPROM or Flash memory), a static random access memory(SRAM), a portable compact disc read-only memory (CD-ROM), a digitalversatile disk (DVD), a memory stick, a floppy disk, a mechanicallyencoded device such as punch-cards or raised structures in a groovehaving instructions recorded thereon, and any suitable combination ofthe foregoing. A computer readable storage medium, as used herein, isnot to be construed as being transitory signals per se, such as radiowaves or other freely propagating electromagnetic waves, electromagneticwaves propagating through a waveguide or other transmission media (e.g.,light pulses passing through a fiber-optic cable), or electrical signalstransmitted through a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network can comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adaptor card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device. Computer readable programinstructions for carrying out operations of the present invention can beassembler instructions, instruction-set-architecture (ISA) instructions,machine instructions, machine dependent instructions, microcode,firmware instructions, state-setting data, configuration data forintegrated circuitry, or either source code or object code written inany combination of one or more programming languages, including anobject oriented programming language such as Smalltalk, C++, or thelike, and procedural programming languages, such as the “C” programminglanguage or similar programming languages. The computer readable programinstructions can execute entirely on the user's computer, partly on theuser's computer, as a stand-alone software package, partly on the user'scomputer and partly on a remote computer or entirely on the remotecomputer or server. In the latter scenario, the remote computer can beconnected to the user's computer through any type of network, includinga local area network (LAN) or a wide area network (WAN), or theconnection can be made to an external computer (for example, through theInternet using an Internet Service Provider). In some embodiments,electronic circuitry including, for example, programmable logiccircuitry, field-programmable gate arrays (FPGA), or programmable logicarrays (PLA) can execute the computer readable program instructions byutilizing state information of the computer readable programinstructions to personalize the electronic circuitry, in order toperform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions. These computer readable programinstructions can be provided to a processor of a general purposecomputer, special purpose computer, or other programmable dataprocessing apparatus to produce a machine, such that the instructions,which execute via the processor of the computer or other programmabledata processing apparatus, create means for implementing thefunctions/acts specified in the flowchart and/or block diagram block orblocks. These computer readable program instructions can also be storedin a computer readable storage medium that can direct a computer, aprogrammable data processing apparatus, and/or other devices to functionin a particular manner, such that the computer readable storage mediumhaving instructions stored therein comprises an article of manufactureincluding instructions which implement aspects of the function/actspecified in the flowchart and/or block diagram block or blocks. Thecomputer readable program instructions can also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational acts to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowcharts and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams can represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the blocks can occur out of theorder noted in the Figures. For example, two blocks shown in successioncan, in fact, be executed substantially concurrently, or the blocks cansometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

While the subject matter has been described above in the general contextof computer-executable instructions of a computer program product thatruns on a computer and/or computers, those skilled in the art willrecognize that this disclosure also can or can be implemented incombination with other program modules. Generally, program modulesinclude routines, programs, components, data structures, etc. thatperform particular tasks and/or implement particular abstract datatypes. Moreover, those skilled in the art will appreciate that theinventive computer-implemented methods can be practiced with othercomputer system configurations, including single-processor ormultiprocessor computer systems, mini-computing devices, mainframecomputers, as well as computers, hand-held computing devices (e.g., PDA,phone), microprocessor-based or programmable consumer or industrialelectronics, and the like. The illustrated aspects can also be practicedin distributed computing environments in which tasks are performed byremote processing devices that are linked through a communicationsnetwork. However, some, if not all aspects of this disclosure can bepracticed on stand-alone computers. In a distributed computingenvironment, program modules can be located in both local and remotememory storage devices.

As used in this application, the terms “component,” “system,”“platform,” “interface,” and the like, can refer to and/or can include acomputer-related entity or an entity related to an operational machinewith one or more specific functionalities. The entities disclosed hereincan be either hardware, a combination of hardware and software,software, or software in execution. For example, a component can be, butis not limited to being, a process running on a processor, a processor,an object, an executable, a thread of execution, a program, and/or acomputer. By way of illustration, both an application running on aserver and the server can be a component. One or more components canreside within a process and/or thread of execution and a component canbe localized on one computer and/or distributed between two or morecomputers. In another example, respective components can execute fromvarious computer readable media having various data structures storedthereon. The components can communicate via local and/or remoteprocesses such as in accordance with a signal having one or more datapackets (e.g., data from one component interacting with anothercomponent in a local system, distributed system, and/or across a networksuch as the Internet with other systems via the signal). As anotherexample, a component can be an apparatus with specific functionalityprovided by mechanical parts operated by electric or electroniccircuitry, which is operated by a software or firmware applicationexecuted by a processor. In such a case, the processor can be internalor external to the apparatus and can execute at least a part of thesoftware or firmware application. As yet another example, a componentcan be an apparatus that provides specific functionality throughelectronic components without mechanical parts, wherein the electroniccomponents can include a processor or other means to execute software orfirmware that confers at least in part the functionality of theelectronic components. In an aspect, a component can emulate anelectronic component via a virtual machine, e.g., within a cloudcomputing system.

In addition, the term “or” is intended to mean an inclusive “or” ratherthan an exclusive “or.” That is, unless specified otherwise, or clearfrom context, “X employs A or B” is intended to mean any of the naturalinclusive permutations. That is, if X employs A; X employs B; or Xemploys both A and B, then “X employs A or B” is satisfied under any ofthe foregoing instances. Moreover, articles “a” and “an” as used in thesubject specification and annexed drawings should generally be construedto mean “one or more” unless specified otherwise or clear from contextto be directed to a singular form. As used herein, the terms “example”and/or “exemplary” are utilized to mean serving as an example, instance,or illustration. For the avoidance of doubt, the subject matterdisclosed herein is not limited by such examples. In addition, anyaspect or design described herein as an “example” and/or “exemplary” isnot necessarily to be construed as preferred or advantageous over otheraspects or designs, nor is it meant to preclude equivalent exemplarystructures and techniques known to those of ordinary skill in the art.

As it is employed in the subject specification, the term “processor” canrefer to substantially any computing processing unit or devicecomprising, but not limited to, single-core processors;single-processors with software multithread execution capability;multi-core processors; multi-core processors with software multithreadexecution capability; multi-core processors with hardware multithreadtechnology; parallel platforms; and parallel platforms with distributedshared memory. Additionally, a processor can refer to an integratedcircuit, an application specific integrated circuit (ASIC), a digitalsignal processor (DSP), a field programmable gate array (FPGA), aprogrammable logic controller (PLC), a complex programmable logic device(CPLD), a discrete gate or transistor logic, discrete hardwarecomponents, or any combination thereof designed to perform the functionsdescribed herein. Further, processors can exploit nano-scalearchitectures such as, but not limited to, molecular and quantum-dotbased transistors, switches and gates, in order to optimize space usageor enhance performance of user equipment. A processor can also beimplemented as a combination of computing processing units. In thisdisclosure, terms such as “store,” “storage,” “data store,” datastorage,” “database,” and substantially any other information storagecomponent relevant to operation and functionality of a component areutilized to refer to “memory components,” entities embodied in a“memory,” or components comprising a memory. It is to be appreciatedthat memory and/or memory components described herein can be eithervolatile memory or nonvolatile memory, or can include both volatile andnonvolatile memory. By way of illustration, and not limitation,nonvolatile memory can include read only memory (ROM), programmable ROM(PROM), electrically programmable ROM (EPROM), electrically erasable ROM(EEPROM), flash memory, or nonvolatile random access memory (RAM) (e.g.,ferroelectric RAM (FeRAM). Volatile memory can include RAM, which canact as external cache memory, for example. By way of illustration andnot limitation, RAM is available in many forms such as synchronous RAM(SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rateSDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), Synchlink DRAM (SLDRAM),direct Rambus RAM (DRRAM), direct Rambus dynamic RAM (DRDRAM), andRambus dynamic RAM (RDRAM). Additionally, the disclosed memorycomponents of systems or computer-implemented methods herein areintended to include, without being limited to including, these and anyother suitable types of memory.

What has been described above include mere examples of systems andcomputer-implemented methods. It is, of course, not possible to describeevery conceivable combination of components or computer-implementedmethods for purposes of describing this disclosure, but one of ordinaryskill in the art can recognize that many further combinations andpermutations of this disclosure are possible. Furthermore, to the extentthat the terms “includes,” “has,” “possesses,” and the like are used inthe detailed description, claims, appendices and drawings such terms areintended to be inclusive in a manner similar to the term “comprising” as“comprising” is interpreted when employed as a transitional word in aclaim.

The descriptions of the various embodiments have been presented forpurposes of illustration, but are not intended to be exhaustive orlimited to the embodiments disclosed. Many modifications and variationswill be apparent to those of ordinary skill in the art without departingfrom the scope and spirit of the described embodiments. The terminologyused herein was chosen to best explain the principles of theembodiments, the practical application or technical improvement overtechnologies found in the marketplace, or to enable others of ordinaryskill in the art to understand the embodiments disclosed herein.

What is claimed is:
 1. A device, comprising: a first qubit; a secondqubit; and a multi-resonant architecture comprising a first resonatorthat couples the first qubit to the second qubit and a second resonatorthat couples the first qubit to the second qubit, wherein the firstqubit is coupled to a first end of the first resonator and the secondqubit is coupled to a second end of the first resonator, wherein thefirst qubit has a first operational frequency, wherein the second qubithas a second operational frequency, and wherein the first resonator hasa first resonant frequency that is less than the first operationalfrequency and the second operational frequency.
 2. The device of claim1, wherein the second resonator has a second resonant frequency that isgreater than the first operational frequency and the second operationalfrequency.
 3. The device of claim 2, wherein the first resonantfrequency is about 3 gigahertz, wherein the second resonant frequency isabout 6 gigahertz, and wherein the first operational frequency and thesecond operational frequency are between 4.5 gigahertz and 5.5gigahertz.
 4. The device of claim 2, wherein the first resonantfrequency, the second resonant frequency, the first operationalfrequency, and the second operational frequency are fixed.
 5. The deviceof claim 1, wherein the first resonator and the second resonator are λ/2resonators, and wherein the first resonator and the second resonator arein parallel.
 6. A device, comprising: a first qubit; a second qubit; anda multi-resonant architecture comprising a resonator, wherein a firstend of the resonator is coupled to the first qubit and to the secondqubit, and wherein a second end of the resonator is coupled to ground,wherein the first qubit has a first operational frequency, wherein thesecond qubit has a second operational frequency, and wherein theresonator has a first harmonic frequency that is less than the firstoperational frequency and the second operational frequency.
 7. Thedevice of claim 6, wherein the resonator has a second harmonic frequencythat is greater than the first operational frequency and the secondoperational frequency.
 8. The device of claim 7, wherein the firstharmonic frequency, the second harmonic frequency, the first operationalfrequency, and the second operational frequency are fixed.
 9. The deviceof claim 6, wherein the resonator is a λ/4 resonator.
 10. The device ofclaim 6, wherein the first harmonic frequency is about 2 gigahertz,wherein the second harmonic frequency is about 6 gigahertz, and whereinthe first operational frequency and the second operational frequency arebetween 4.5 gigahertz and 5.5 gigahertz.
 11. An apparatus, comprising: afirst transmon qubit having a first operational frequency; a secondtransmon qubit having a second operational frequency; and amulti-resonant architecture that couples the first transmon qubit to thesecond transmon qubit, wherein the multi-resonant architecture has afirst resonant frequency less than the first operational frequency andthe second operational frequency.
 12. The apparatus of claim 11, whereinthe multi-resonant architecture has a second resonant frequency greaterthan the first operational frequency and the second operationalfrequency, and wherein the multi-resonant architecture comprises a firstλ/2 resonator coupled to the first transmon qubit and to the secondtransmon qubit and comprises a second λ/2 resonator coupled to the firsttransmon qubit and to the second transmon qubit, wherein the first λ/2resonator and the second λ/2 resonator are in parallel, wherein thefirst λ/2 resonator exhibits the first resonant frequency, and whereinthe second λ/2 resonator exhibits the second resonant frequency.
 13. Theapparatus of claim 12, wherein the first resonant frequency is about 3gigahertz, and wherein the second resonant frequency is about 6gigahertz.
 14. The apparatus of claim 11, wherein the multi-resonantarchitecture comprises a λ/4 resonator, wherein a first end of the λ/4resonator is coupled between coupling capacitors of the first transmonqubit and the second transmon qubit, wherein a second end of the λ/4resonator is shorted to ground, wherein a first harmonic of the λ/4resonator is the first resonant frequency, and wherein a second harmonicof the λ/4 resonator is the second resonant frequency.